Zero-density estimates for Epstein zeta functions

We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients in the vertical strip $\sigma_1 < \Re s < \sigma_2$, where $1/2 < \sigma_1 < \sigma_2 < 1$. When the class number of the quadratic form is bigger than 1, Voronin gives a lower bound and Lee gives an asymptotic formula for the number of zeros. In this paper, we improve their results by providing a new upper bound for the error term.